Mr Cundy's announcement coincided with the police releasing images showing the charred wreckage inside the tower, which show flats reduced to ash and rubble.
However, Mr Cundy went on to describe the search operation as "hugely complex" and said: "The bad reality is, as I've said before, is that due to the intensity of the fire and the devastation within Grenfell Tower, we may not be able to identify everybody that died".
Adele also joined the firefighters in a minute's silence for the victims of the fire, observed across the UK.
Mr Cundy said the death toll of 79 may still change, but not as significantly as it has in recent days.
He fought back tears as he told reporters at New Scotland Yard about the scene inside the tower.
"It is incredibly hard to describe the devastation in some parts of the building", he continued, his voice breaking.More news: Woman picks up deadly rattlesnake thinking it was a toy
5 people who were missing have turned up safe.
Residents will be given £500 in cash followed by a bank payment for the rest from Monday with the money coming from the £5m fund announced by British Prime Minister Theresa May last week.
The announcement comes after the council began a review over the weekend of its housing blocks.
This letter sent by Kensington and Chelsea's council leader, Nick Paget-Brown, to residents in 2014 will absolutely infuriate you. Because of the intensity of the blaze some may never be identified.
Khadija Khalloufi, who lived in Grenfell Tower, had not been seen since a huge blaze destroyed the 24-storey tower block in Notting Hill.
"But this year, it's hard not to feel the very dark national mood", she said, inviting the British to tap into their resources, in their history, to respond to the "succession of bad tragedies" Month: three bloody attacks and the burning of the Grenfell Tower in London. "For three weeks I have to come here with the children and wash ourselves?"
It was reported that the material used in the cladding covering Grenfell was Reynobond PE - a cheaper, more flammable version of two available options.More news: Change of Prime Minister 'not on the agenda' - Chris Grayling
He said: "We have been from the top to the bottom of Grenfell Tower".
However John Cowley, managing director of CEP Architectural Facades, which provided the rainscreen panels for Grenfell Tower's sub-contractor Harley Facades Ltd, said that was untrue.
"Current building regulations allow its use in both low-rise and high-rise structures". She was going around talking to people.
The rest are missing presumed dead and Mr Cundy said "the very bad reality is that we may not be able to identify all those that died" inside what is now a blackened and charred shell.
He said police had received some 70 pictures and videos of the fire from the public and urged them to send more as officers investigate the blaze.
So far, 79 people are feared dead.
"So we opened the door to her and then she took her sunglasses off and said: "Hi, I'm Adele".More news: Vegas holds trade cards during expansion draft window
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